64 problems

Let $M$ be a spin manifold with a positive Ricci curvature metric, and let its Witten genus be viewed as an invariant in the appropriate ring of modular forms. Stolz's conjecture.…

Yau's conjecture. The manifold $X$ is projective and rationally connected.

Let $M$ be a Riemannian $4k$-manifold with positive Ricci curvature that admits a string structure, meaning that the classifying map of its tangent bundle lifts from…

Wilhelm's conjecture. Every such submersion satisfies

Let $\pi\colon (M^{n+k},\textsl g)\to (B^n,b)$ be a Riemannian submersion. Suppose that $(M,\textsl g)$ is compact and has positive sectional curvature. Petersen–Wilhelm's conjectu…

Yau's conjecture. If $X$ has a Kähler metric with positive holomorphic sectional curvature, then $X$ is a projective and rationally connected manifold.

Let $M^n$ be a connected positively curved manifold, and let $b_{2i}(M)=\dim H_{2i}(M;\mathbb{F})$ denote its even-degree Betti numbers over a coefficient field. Gromov's linear Be…

Positive 2-curvature conjecture. $M$ admits a Riemannian metric of positive $2$-curvature if and only if

Let $(M, \langle\;, \: \rangle)$ be a complete Riemannian manifold with quasi-positive curvature, meaning that it has non-negative sectional curvature and there is a point at which…

Klingenberg–Sakai conjecture. There are only finitely many positively curved manifolds in a given homotopy type.

Let $M$ be a positively curved manifold, and let $\pi_1(M)$ denote its fundamental group. Chern's conjecture. Every abelian subgroup of $\pi_1(M)$ is cyclic. This is the positive-c…

Assume that $M$ is a compact simply connected 5-manifold with a metric of non-negative sectional curvature. The known models are $\mathbb{S}^5$,…

Let $M$ be an irreducible compact Kähler manifold with positive orthogonal bisectional curvature. Projective-space conjecture. Is $M$ necessarily biholomorphic to…

Let $M$ be a two-dimensional complete manifold, and fix an origin $O$. Say that its curvature is greater than or equal to an SL-bifurcator when it satisfies the comparison conditio…

Let $M$ be a compact spin manifold with positive Ricci curvature and vanishing first Pontryagin class … The Hoehn–Stolz conjecture. The Witten genus of $M$ vanishes: … Here $D^L$ i…

Let $M^n$ be a manifold and let $\delta$-pinched metrics on $M^n$ mean metrics whose sectional curvatures satisfy the pinching condition determined by $\delta$. Consider the inject…

The existence conjecture. Every simply connected Riemannian manifold $(M,\mathtt{g})$ with positive (respectively, non-negative) sectional curvature admits a codimension-one singul…

Let $\mathcal{O}_{p,q}^{a,b}$ be an Eschenburg orbifold obtained from a positively curved Eschenburg $7$-manifold by an isometric circle action. Florit–Ziller conjecture. No positi…

Let $M$ be a simply connected manifold equipped with a non-negatively curved metric, and suppose that the metric has positive sectional curvature at one point. Deformation conjectu…

Ramachandran–Wolfson fill-radius conjecture. Uniformly 2-positive Ricci curvature and uniformly positive isotropic curvature should each imply a bound on the fill radius.

Let $k$ be a positive integer, and let $S^{2k}$ or $S^{2k-1}$ be an even- or odd-dimensional sphere equipped with a Finsler metric. Katok–Anosov conjecture. Every Finsler metric on…

Let $\rho^+_{\beta_{\mathrm q}}(A)$ be the area distribution of positive-curvature JT gravity coupled to conformal matter with $c\leq0$, and let $\nu$ be the corresponding critical…

Let $(M^n,g_0)$ be a complete WPIC1 manifold with $n\geq 3$. WPIC1 Ricci-flow existence conjecture. There exists a complete WPIC1 Ricci flow $g(t)$ on $M$ for $t\in[0,T)$, for some…

Let $M$ be an open Kähler manifold with positive holomorphic bisectional curvature. Yau's conjecture. $M$ is biholomorphic to $\mathbb{C}^n$. This is the Kähler analogue of the exp…

Let $M$ be an open, complete, noncompact manifold without boundary whose curvature is PIC1. PIC1 open-manifold topology conjecture. $M$ is diffeomorphic to Euclidean space. This co…